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## geometrical optics

### (3 points)6. Series 32. Year - 1. selfenlightment

We illuminate a mirror at an angle of $\alpha = 15\mathrm{\dg }$ with respect to the normal. We want the light to travel directly back to the source. For doing so, we can use a glass prism with an index of refraction $n = 1,8$. Find the angle $\eta $ as a function of $\alpha $ and $n$ (see the figure). The prism is placed into the air with an index of refraction $n_0$.

**Hint:**
\[\begin{align*}
\sin \(x + y\) &= \sin x \cos y + \cos x \sin y , \\

\cos \(x + y\) &= \cos x \cos y - \sin x \sin y , \\

\sin x + \sin y &= 2\sin \(\frac {x + y}{2}\)\cos \(\frac {x - y}{2}\) , \\

\cos x + \cos y &= 2\cos \(\frac {x + y}{2}\)\cos \(\frac {x - y}{2}\) .
\end {align*}\]

Karel saw Danka's task.

### (6 points)2. Series 32. Year - 3. physics trophy

Danka won the annual Derivative Bee and she obtained a statuette made of transparent material as a reward. This statuette is made in shape of a cube prism with an edge of $a = 5$ cm and height of $h \leq a$. No matter what angle she looks at the prism, she can only see the reflection on the side walls but not through it. What is the index of refraction of the material? The prism is placed in air.

Michal K. was charmed by a statuette.

### (3 points)5. Series 31. Year - 2. death rays on the glass

A light ray falls on a glass plate with an absolute reflective index $n = 1,5$. Determine its angle of incidence $\alpha _1$ if the reflected ray forms an angle $60 \dg$ with the refracted ray. The board is stored in the air.

Danka likes solvine more problems simultaneously.

### (3 points)2. Series 31. Year - 2. solar power plant

The solar constant, or more accurately the solar irradiance, is the influx of energy coming from the Sun at the distance where Earth is. It technically doesn't have a constant value, but let's suppose it is approximately $P = 1{,}370\,\mathrm{W\cdot m^{-2}}$. Also, suppose that Earth's orbit is circular and its axis of rotation is tilted with respect to the normal of the orbital plane by $23.5\dg $. What would be the maximum power captured by a solar panel of area $S= 1\,\mathrm{m^2}$ at the summer and winter solstice, if the panel lies flat on the ground in Prague (latitude $50\dg $ N)? Ignore the effects of any obstructions or the atmosphere.

Karel watched Crash Course Astronomy

### (6 points)2. Series 31. Year - 3. observing

What fraction of a spherical planet's surface cannot be seen from the stationary orbit above the planet? (A stationary orbit is one where the satellite stays fixed above a certain point on the planet.) The density of the planet is $\rho $ and its rotation period is $T$.

Filip went through the unseen competition problems.

### (3 points)1. Series 31. Year - 2. backup NAS(A)

Consider an optical switch (transfer speed $10 \mathrm{Gb s^{-1}}$), whose output (after any necessary amplification) is used to illuminate the Moon. Thanks to the mirrors left behind by the Apollo mission, the signal comes back and can be used (after any necessary amplification) as an input to the switch. If we make sure the switch works reliably the transmitted data will circle in the system indefinitely. Thus we acquire a memory. What is its maximum capacity? Ignore any delays caused by the processing of the signal and any headers of the data.

Michal combined pingf and Laufzeitspeicher

### (8 points)5. Series 30. Year - P. glasses

Describe the imaging system of a microscope (consisting of two convex lenses) and that of a Keplerian telescope. Explain the difference in function and construction of a microscope and a telescope and sketch the rays passing through the systems. How can we usefully define magnification for these optical systems? Derive the equations for magnification.

Kuba finally understood, how it all works!

### (9 points)4. Series 30. Year - 5. weird atmosphere

Have you ever seen such a weird atmosphere? Up to a certain height the speed of light inside it is constant, $v_{0}$, but from that certain height the speed of light starts increasing linearly as $v(Δh)=v_{0}+kΔh$. At one point, exactly at the height where the speed of light starts changing, light beams are sent upward in all directions. Show that all these beams move along circular arcs and determine the radii of these arcs. Also find out the distance between the place where the the light was emitted and the point where the beams return to the original height.

Jakub wanted to know what it would be like to swim under ice.

### (2 points)6. Series 29. Year - 2. Optometric

Pikos' friend wears glasses. When she puts them on, her eyes seem to be smaller. Is she shortsighted or farsighted? Justify your answer.

### (5 points)2. Series 29. Year - 4. mirrorception

Consider an optical system composed of three semitransparent mirrors placed behind each other along one axis. Every mirror by itself reflects half of incident light and lets the other half pass. Determine what fraction of light passes through the system of mirrors.

**Bonus:** Solve the problem for $n$ such mirrors.

Karel se prohlížel v zrcadle.